If I move an Elastic IP from one EC2 instance to another, does it happen instantly?

I am asking this question, as I have a situation where I have a website hosted within Elastic Beanstalk without a load balancer. Therefore my Route53 is configured so that my domain is pointing directly to the Elastic IP of my EC2 instance. Thus, I can't do the normal CNAME swap as suggested.

Now, I have looked at the pricing of the Elastic Load Balancer and don't have a requirement for it because my site doesn't receive enough traffic to warrant it. However, I do have a requirement to be able to release new versions of the site without down time and without adversely affecting users by having to wait for settings to propagate 'through the internet' so it were.

I was told that it is possible to reassign Elastic IP addresses, so what I thought was that I could do the following, when releasing a new version of my website:

Create a new EC2 instance

Deploy the new version of the application to the new EC2 instance

Check that the new version is working on the new EC2

Perform a swap of the Elastic IP addresses to make the new version live

Shut down the old EC2 instance

I'm concerned that switching the Elastic IP addresses could take time to begin to direct users to the new instance.

Also, is performing a Elastic IP swap this quite a 'standard' way of releasing new versions of websites within AWS?